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Chapter 0: Problem 109

Simplify each exponential expression. Assume that variables represent nonzero real numbers. $$\left(2 x^{-3} y z^{-6}\right)(2 x)^{-5}$$

Short Answer

Expert verified
The simplified form of the given expression is \(2^{-4} \cdot x^{-8} \cdot y \cdot z^{-6}\).

Step by step solution

01

Apply Product of Powers Rule

Firstly, distribute the power of -5 across the base of 2x in the second term. According to the product of powers rule, for all real numbers \(a\), \(b\) and \(n\), the formula is \((ab)^n = a^n \cdot b^n\). So, \((2x)^{-5}\)can be expressed as \(2^{-5} \cdot x^{-5}\). The new expression becomes: \((2x^{-3} yz^{-6}) \cdot 2^{-5} \cdot x^{-5}\).
02

Multiply coefficients and variables separately

Next, separate the coefficients (numbers) from the variables and multiply them separately. This will result in \(2 \cdot 2^{-5}\) for coefficients and \(x^{-3} \cdot x^{-5}\) for x-variables which leaves us with \(y\) and \(z^{-6}\) to be included in the final expression. By doing this we get: \(2 \cdot 2^{-5} \cdot x^{-3} \cdot x^{-5} yz^{-6}\).
03

Simplify coefficients and variables give final result

Now simplify each part of above expression. The coefficient \(2 \cdot 2^{-5} = 2^{-4}\) and \(x^{-3} \cdot x^{-5} = x^{-8}\), because according to power rule, powers should be added while multiplying. So, the final result after simplification, the expression becomes: \(2^{-4} \cdot x^{-8} \cdot y \cdot z^{-6}\).

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